The single-engine Mooney M20C belonged to one of the boys' parents, and the pair was trying to run away from their homes, said Cindy Beavers, spokeswoman for the San Bernardino County Sheriff's Department.
"There was a disagreement between the boys and their parents, and (the boys) decided to run away," she said.
The boys took the plane from Big Bear Airport Sunday afternoon, authorities said. The boys bypassed three levels of security before taking off, taking the keys to get into the airport and the hangar, as well as the keys for the aircraft itself, said Garry Dokter, general manager of the airport, the Los Angeles Times reports. The airport has no control tower and pilots are not required to check in with anyone before taking off, said Garry Dokter, the airport's general manager.
Two pilots saw the boys taxiing and called the plane's owner and the Sheriff's Department.
The plane crashed in Joshua Tree National Park between 5:30 p.m. and 6 p.m. The boys suffered minor injuries and were treated at the scene by rangers, according to park officials.
Officials said they didn't know why the plane went down but speculated it ran out of gas.
Although neither boy has a pilot's license, the one who piloted the plane flew regularly with his father, Dokter said.
The boys were arrested for investigation of airplane theft and burglary, Beavers said. They were booked into San Bernardino County Juvenile Hall on Monday.